William Brown - also known as William Lindsay - died at Polmont Young Offenders' Institution on October 7 2018, aged 16, days after being sent there on remand.
The teenager's mother, Christine Lindsay, believes her son was not safe in his cell and said it is her duty to "cry out for justice".
She said Scottish politicians should "hang their heads in shame" over not taking more action to tackle suicide rates in the country's prisons.
The Crown Office said in September 2019 that it would not bring charges against the Scottish Prison Service (SPS) over William's death.
The family's lawyer Aamer Anwar said SPS and NHS Forth Valley will be served with a summons on Tuesday October 26.
In a statement on behalf of the family, Mr Anwar said: "William was in the 'high risk of suicide' category.
"In the period 2012 to 2018, William had significant involvement with mental health services and was the subject of numerous assessments."
The statement added: "Despite this information, we know he was removed from the observation cell and William was described as 'being safe and secure', and consequently was assessed as 'no apparent risk' and removed from the prison 'talk to me strategy'.
"It is alleged he was not treated according to the standard operating procedures as a child should have been."
On Saturday October 6, the family understand that William was in his cell on his own for long periods of time.
He was found dead the next day.
Mr Anwar said William's family "no longer have any trust in the Scottish Government, the Scottish Prison Service or an FAI to deliver justice or the truth."
He added: "Corporate homicide is, of course, the correct legal term, but William's mother believes that what happened to her son can only be described as 'state-sanctioned murder'."
After learning SPS would not be prosecuted, the family submitted a victims right to review (VRR) - a scheme which enables victims to seek a review of why the Crown Office Procurator Fiscal Service has decided to prosecute or not prosecute.
The VRR said: "The family of William Brown or Lindsay argue that SPS were under a duty to take reasonable care to prevent him while in custody from taking his own life deliberately.
"SPS failed to do so, and in what can only be characterised as a gross breach of their relevant duty of care, caused or materially contributed to his death."
The review continued: "William was an obvious high suicide risk.
"His suicide on October 7 2018, was therefore a foreseeable consequence of SPS's failure of duty, which occurred when he was placed in a cell in circumstances that provided him with the opportunity to carry out that act.
"In these circumstances, as recognised in case law referred to in this VRR, SPS must be held accountable and indicted for the crime of corporate homicide under The Corporate Manslaughter and Corporate Homicide Act 2007."
Forth Valley have been contacted for comment.